Math Conjectures

Brennan conjecture

OPENMajorConjectureProposed 1978 · Full conjecture

Canonical statement

Let dAdA be Lebesgue area measure, let Ω⊊C\Omega\subsetneq\mathbb C be simply connected, and let f:D→Ωf:\mathbb D\to\Omega be conformal. For every real ss with −2<s<2/3-2<s<2/3,
∫D∣f′(z)∣s dA(z)<∞. \int_{\mathbb D}|f'(z)|^s\,dA(z)<\infty .
Equivalently, for a conformal ϕ:Ω→D\phi:\Omega\to\mathbb D, ϕ′∈Lp(Ω,dA)\phi'\in L^p(\Omega,dA) for every 4/3<p<44/3<p<4.
View source LaTeX
Let \(dA\) be Lebesgue area measure, let \(\Omega\subsetneq\mathbb C\) be simply connected, and let \(f:\mathbb D\to\Omega\) be conformal. For every real \(s\) with \(-2<s<2/3\), \[ \int_{\mathbb D}|f'(z)|^s\,dA(z)<\infty . \] Equivalently, for a conformal \(\phi:\Omega\to\mathbb D\), \(\phi'\in L^p(\Omega,dA)\) for every \(4/3<p<4\).
ANAL-FA-011:[Brennan1978ConformalMappings][HedenmalmShimorin2005WeightedBergman][BeliaevSmirnov2010IntegralMeans]
Analysis and functional analysis

Notes

Brennan's conjecture, posed in 1978, concerns the integrability of derivatives of conformal maps. If ff maps the unit disc conformally onto a simply connected domain Ω⊊C\Omega\subsetneq\mathbb C, the conjecture predicts ∫D∣f′∣s dA<∞\int_{\mathbb D}|f'|^s\,dA<\infty for every −2<s<2/3-2<s<2/3; equivalently, the derivative of a conformal map ϕ:Ω→D\phi:\Omega\to\mathbb D lies in Lp(Ω)L^p(\Omega) for all 4/3<p<44/3<p<4 [Brennan1978ConformalMappings]. Classical distortion estimates give part of this range, and examples show that the conjectured range would be sharp.

The problem is closely tied to the universal integral means spectrum of conformal maps. Hedenmalm and Shimorin improved the known exponent range through estimates in weighted Bergman spaces [HedenmalmShimorin2005WeightedBergman], and Beliaev and Smirnov studied the integral means spectrum via harmonic measure and SLE techniques [BeliaevSmirnov2010IntegralMeans].

Despite substantial improvements on both endpoint-adjacent ranges, the complete interval remains unproved for arbitrary simply connected domains, and the conjecture is still open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.